2.1 Cdg categories and functors

Recall that a curved differential graded category $\mathscr {C}$ (cdg category for short) is a triple $(\mathscr {C},d,h)$ where $\mathscr {C}$ is a category enriched over $\mathbb {Z}/2$ -graded vector spaces, d is an assignment of a degree $1$ operator (called differential) $d_{x,y}:\text {Hom}_{\mathscr {C}}(x,y)\to \text {Hom}_{\mathscr {C}}(x,y)[1] $ for every pair of objects x and y in $\mathsf {Ob}(\mathscr {C})$ and h is the assignment of a degree $0$ endomorphism $h_x\in \mathrm {End}_{\mathscr {C}}(x)$ (called curvature) for every object $x\in \mathsf {Ob}(\mathscr {C})$ . This structure d and h must satisfy the following, d is a derivation for the composition, $d_{x,x}(h_x)=0$ and $d_{x,y}^2(f)=h_y\circ f-f\circ h_x$ for all $f\in \text {Hom}_{\mathscr {C}}(x,y)$ . Here are some examples that will be important to us.

Let $\mathscr {C}$ and $\mathscr {C}'$ be two cdg categories. A functor $F:\mathscr {C}\to \mathscr {C}'$ of categories enriched over graded vector spaces is called a quasi-cdg functor if $F_{x,y}\circ d_{x,y}=d_{F(x),F(y)}\circ F_{x,y}$ for all objects x and y. If in addition we have $F(h_x)=h_{F(x)}$ for all objects x then we call F a cdg functor.

Cdg (quasi-)modules over a cdg category $\mathscr {C}$ are by definition (quasi) cdg functors $F:\mathscr {C}\to Com_{cdg}(k)$ . If F is a quasi-cdg functor we denote by $F[n]$ the quasi-cdg functor obtained by composing $[n]\circ F$ . The collection of quasi-modules themselves form a cdg category that we will denote by $mod^{qdg}-\mathscr {C}$ . The objects are quasi-modules. A degree n morphism $\alpha \in \text {Hom}_{mod^{qdg}-\mathscr {C}}^n(F,F')$ is a natural transformation $\alpha :F\implies F'[n]$ between functors of categories enriched over graded vector spaces. The differentials $d_{F,F'}$ are given by $d_{F,F'}(\alpha )_x:=\delta _{F'(x)}\circ \alpha _x-(-1)^n\alpha _x\circ \delta _{F(x)}$ and the curvatures are by definition $h_F(x):=h_{F(x)}-F(h_x)$ . The full subcategory of modules inside the cdg category of quasi-modules forms a dg-category which we denote by $mod^{cdg}-\mathscr {C}$ . If $\mathscr {C}$ is a dg category we will write $mod^{dg}-\mathscr {C}$ for the dg category $mod^{cdg}-\mathscr {C}$ .

2.2 Two kinds of Hochschild homology

Let $\mathscr {C}$ be a cdg category. We define the Hochschild complex of the first kind $C_{\bullet }(\mathscr {C})$ as follows. As a $\mathbb {Z}/2$ -graded vector space it is obtained by taking the direct sum along the diagonal of a bigraded vector space with one grading by $\mathbb {Z}$ and one by $\mathbb {Z}/2$ . The $\mathbb {Z}/2$ -graded vector space in degree n is

$$ \begin{align*} C_n(\mathscr{C})=\bigoplus_{X_0,...,X_n\in \mathsf{Ob}(\mathscr{C})}\text{Hom}(X_0,X_1)\otimes\text{Hom}(X_1,X_2)\otimes...\otimes\text{Hom}(X_n,X_0). \end{align*} $$

The differential is the sum of three differentials on $C_{\bullet }(\mathscr {C})$ :

$$ \begin{align*} d_0\big(a_0[a_1|\cdots |a_k]\big)&=\sum_{i=0}^k(-1)^{|a_0|+...+|a_i|+i}a_0[\cdots|a_i|h|a_{i+1}|\cdots]\\ d_1\big(a_0[a_1|\cdots |a_k]\big)&=\sum_{i=0}^k(-1)^{|a_0|+...+|a_{i-1}|+i}a_0[\cdots|da_i|\cdots]\\ d_2\big(a_0[a_1|\cdots |a_k]\big)&=(-1)^{|a_0|}a_0a_1[a_2|\cdots|a_k]\\ &+\sum_{i=1}^{k-1}(-1)^{|a_0|+...+|a_i|+i}a_0[\cdots|a_ia_{i+1}|\cdots]\\ &+(-1)^{1+(|a_k|+1)(|a_0|+...+|a_{k-1}|+k-1)}a_ka_0[a_1|\cdots|a_{k-1}] \end{align*} $$

The Hochschild homology $HH_{\bullet }(\mathscr {C})$ is the object in $D_{Z/2}(\mathbb {C})$ represented by $C_{\bullet }(\mathscr {C})$ . Although Hochschild homology is an important invariant of dg categories it turns out that for general cdg categories it is not very useful. For example if $h\neq 0$ in example 3 then $HH_{\bullet }(A,0,h)=0$ ([Reference Căldăraru and Tu4], theorem 4.2). There is a notion of Hochschild homology of the second kind which in some ways is a better invariant for cdg categories. The Hochschild complex of the second kind associated to a cdg category $\mathscr {C}$ , denoted $C_{\bullet }^{II}(\mathscr {C})$ , is defined from the same bigraded vector space as for the first kind but we take direct products along the diagonal instead of direct sum. The differential is given by the same formulas as for the first kind.

There is a canonical map $C_{\bullet }(\mathscr {C})\to C_{\bullet }^{II}(\mathscr {C})$ and it is an interesting question when it is an isomorphism. Both Hochschild complexes of the first and second kind are functorial with respect to cdg functors as defined in this text.

2.3 Pseudo equivalences and projective modules

Let $\mathscr {D}$ be a cdg category. We say that an object $x\in \mathsf {Ob}(\mathscr {D})$ is the direct sum of two objects $y\oplus z$ if there are degree zero morphisms $\iota _y:y\to x$ and $\iota _z:z\to x$ which induce isomorphisms of graded groups $\text {Hom}(y,w)\times \text {Hom}(z,w)\cong \text {Hom}(x,w)$ and we require that $d(\iota _y)=0$ and $d(\iota _z)=0$ .

If $y\in \mathsf {Ob}(\mathscr {D})$ and $\tau \in \text {Hom}^1(y,y)$ we say that an object $x\in \mathsf {Ob}(\mathscr {D})$ is a twist of y by $\tau $ (and we write $x=y(\tau )$ ) if there are degree $0$ morphisms $i:y\to x$ and $j:x\to y$ such that $ij=\mathrm {id}_x$ , $ji=\mathrm {id}_y$ and $jd(i)=\tau $ .

Finally we call an object $x\in \mathsf {Ob}(\mathscr {D})$ a degree n shift of another object $y\in \mathsf {Ob}(\mathscr {D})$ if there are morphisms $i:y\to x$ and $j:x\to y$ of degree n and $-n$ respectively such that $ij=\mathrm {id}_x$ , $ji=\mathrm {id}_y$ , $d(i)=0$ and $d(j)=0$ .

We call a cdg functor $F:\mathscr {C}\to \mathscr {D}$ pseudo equivalence if it is fully faithful and any object in $\mathscr {D}$ can be obtained from objects in the image of F in finitely many steps by taking direct sums, twisting, shifting or passing to direct summands.

For a cdg category $\mathscr {C}$ we denote by $\mathscr {C}^\#$ the underlying category enriched in graded vector spaces. The category of graded modules on $\mathscr {C}$ (i.e. graded functors from $\mathscr {C}^\#$ to graded vector spaces) is an abelian category. The projective modules are precisely those that are direct summands of a direct sum of representable modules. We say that a module P is finitely generated projective if it is a direct summand of a finite direct sum of representable functors. Let’s denote by $mod^{qdg}_{fgp}-\mathscr {C}$ the full subcategory of $mod^{qdg}-\mathscr {C}$ consisting of quasi-modules that are finitely generated projective as $\mathscr {C}^\#$ -modules. Let $mod^{cdg}_{fgp}-\mathscr {C}$ denote the full subcategory of $mod^{qdg}_{fgp}-\mathscr {C}$ whose curvature vanishes. Let $\mathscr {C}=(A,0,h)$ and $\mathscr {C}'=(A,0,-h)$ . We have a Yoneda functor $R:\mathscr {C}'\to mod^{qdg}_{fgp}-\mathscr {C}$ which is a cdg functor. We also have the inclusion functor $I:mod^{cdg}_{fgp}-\mathscr {C}\to mod^{qdg}_{fgp}-\mathscr {C}$ .

2.4 Quasi-functors between dg categories

Let $\mathscr {C}$ be a dg category. The categories $Z^0(\mathscr {C})$ or $H^0(\mathscr {C})$ are obtained from $\mathscr {C}$ by taking degree zero cycles or by taking zeroth homology of all the hom-complexes in $\mathscr {C}$ respectively. Recall that the module category $mod^{dg}-\mathscr {C}$ is again a dg category and $H^0(mod^{dg}-\mathscr {C})$ has a standard triangulated structure. If we localize $H^0(mod^{dg}-\mathscr {C})$ with respect to quasi-isomorphisms we get the derived category $D(mod^{dg}-\mathscr {C})$ . Using Drinfeld quotients ([Reference Drinfeld6]) we get a dg enhancement of $D(mod^{dg}-\mathscr {C})$ , which we denote $\mathscr {D}(mod^{dg}-\mathscr {C})$ whose objects are the same as $mod^{dg}-\mathscr {C}$ . We call an object in $mod^{dg}-\mathscr {C}$ perfect if it lies in the smallest thick (meaning closed under passage to a direct summand) triangulated subcategory of $D(mod^{dg}-\mathscr {C})$ which contains the image of the Yoneda functor $H^0(\mathscr {C})\to D(mod^{dg}-\mathscr {C})$ . Let $\text {perf}(\mathscr {C})\subset \mathscr {D}(mod^{dg}-\mathscr {C})$ be the full sub dg category on the perfect objects in $mod^{dg}-\mathscr {C}$ . If $\mathscr {C}$ is pre-triangulated (meaning $H^0(\mathscr {C})$ is triangulated in a way compatible with the Yoneda embedding $H^0(\mathscr {C})\to H^0(mod^{dg}-\mathscr {C})$ ) and idempotent complete then the Yoneda functor $\mathscr {C}\to \text {perf}(\mathscr {C})$ is a quasi-equivalence. If $\mathscr {C}$ and $\mathscr {C}'$ are two dg categories, both pre-triangulated then we define a quasi-functor from $\mathscr {C}$ to $\mathscr {C}'$ to mean a dg functor $F:\mathscr {C}\to \text {perf}(\mathscr {C}')$ . We will denote a quasi-functor by a dashed arrow $F:\mathscr {C}\rightarrow \mathscr {C}'$ . By a theorem of Keller, (theorem 2.4 in [Reference Keller9]), if $\mathscr {C}$ is a dg category then the Yoneda functor induces an isomorphism on Hochschild homology $C_{\bullet }(\mathscr {C})\to C_{\bullet } (\text {perf}(\mathscr {C}))$ and therefore any quasi-functor $F:\mathscr {C}\dashrightarrow \mathscr {C}'$ induces a well-defined map in $D(k)$ , $C_{\bullet }(\mathscr {C})\to C_{\bullet }(\mathscr {C}')$ .

2.5 Presheaves of cdg categories

We will also need presheaf versions of the Hochschild complexes. A presheaf of cdg categories $\underline {\mathscr {E}}$ on a scheme X is a contravariant functor from the category of open subsets of X and inclusions of such to the category of small cdg categories with cdg functors as morphisms. If for all U, $\underline {\mathscr {E}}(U)$ has only one object we call it a presheaf of cdg algebras and if $U\mapsto \underline {\mathscr {E}}(U)$ is a sheaf we call it a sheaf of cdg algebras. If for all U, $\underline {\mathscr {E}}(U)$ is a dg category then we call it presheaf of dg categories. Here is an example that will be important to us later.

A morphism of presheaves of cdg categories $\varphi :\underline {\mathscr {E}}\to \underline {\mathscr {E}}'$ is the data a cdg functor $\varphi _U:\underline {\mathscr {E}}(U)\to \underline {\mathscr {E}}'(U)$ for each open set $U\subset X$ such that whenever $V\subset U\subset X$ the following diagram commutes on the nose

If $\underline {\mathscr {E}}$ is a presheaf of cdg categories we define $\underline {C}_{\bullet }^{II}(\underline {\mathscr {E}})$ to be the presheaf of $\mathbb {Z}/2$ -graded chain complexes

$$ \begin{align*} \underline{C}_{\bullet}^{II}(\underline{\mathscr{E}})(U):= C_{\bullet}^{II}(\underline{\mathscr{E}}(U)). \end{align*} $$

Similarly one defines $\underline {C}_{\bullet }(\underline {\mathscr {E}})$ . Any morphism $\varphi :\underline {\mathscr {C}}\to \underline {\mathscr {D}}$ of presheaves of cdg categories gives rise to a map of presheaves of complexes $\varphi _{\ast }:\underline {C}_{\bullet }^{?}(\underline {\mathscr {C}})\to \underline {C}_{\bullet }^?(\underline {\mathscr {D}})$ (where $?=I, II$ ).

We will need a weaker notion of morphism between presheaves of cdg categories where the cdg functors are only asked to commute with the restriction functors up to compatible natural isomorphisms. Here is the exact definition. Let $\underline {\mathscr {C}}$ and $\underline {\mathscr {D}}$ be two presheaves of cdg categories on a scheme X.

Definition 10. A lax morphism $\varphi :\underline {\mathscr {C}}\to \underline {\mathscr {D}}$ is a collection of cdg functors $\{\varphi _U:\underline {\mathscr {C}}(U)\to \underline {\mathscr {D}}(U)| \ U\subset X\text { open}\}$ together with isomorphisms of functors $\{\alpha _{UV}:\varphi _{V}\circ \text {Res}_{UV}\implies \text {Res}_{UV}\circ \varphi _U| \ V\subset U\subset X \text { open }\}$ which satisfies the following cocycle condition, for all $c\in \mathsf {Ob}(\underline {\mathscr {C}})$ and all $W\subset V\subset U$ we have

$$ \begin{align*}\alpha_{UW,c}= \text{Res}_{VW}(\alpha_{UV,c})\circ \alpha_{VW,\text{Res}_{UV}(c)}\end{align*} $$

Suppose $X=\cup _{i=1,...,n}U_i$ is a finite union of affine schemes. We will write ${\check{C}}(\mathscr {U},\underline {C}_{\bullet }^{?}(\underline {\mathscr {C}}))$ for the total complex obtained by taking the total complex of the bicomplex ${\check{C}}^p(\mathscr {U},\underline {C}_{-q}^{?}(\underline {\mathscr {C}}))$ where $?=I,II$ (see Section ?? for our conventions on the differentials in this situation). We will see that lax morphisms induce chain maps on Cech-Hochschild complexes associated to a presheaf of cdg categories as above. To understand this we first introduce some notation. Let $(\varphi ,\alpha )$ be a lax morphism. If $I=\{i_0<...<i_p\}$ , $J=\{j_1,...,j_q\}$ and $J\cap I=\emptyset $ then we set $J_s:=\{j_1,j_2,...,j_s,i_0,...,i_p\}$ for $0\leq s\leq q$ . With this notation we define the following maps $h^{q}:{\check{C}}(\mathscr {U},\underline {C}^{?}_{\bullet }(\underline {\mathscr {C}}))\to {\check{C}}(\mathscr {U},\underline {C}^{?}_{\bullet }(\underline {\mathscr {D}}))$ by

(4) $$ \begin{align}&C_k(\underline{\mathscr{C}}(U_{I}))\ni a_0[a_1|\cdots |a_k]\mapsto \nonumber\\ &\quad\sum_J (-1)^{\epsilon+\sigma(I,J)+q}\alpha\circ\varphi(a_0|_{J_q})\Big[\cdots\Big|\alpha^{-1}\Big|\varphi(a_{l_1+1}|_{J_{q-1}})|_{J_q}\Big|\cdots\Big|\alpha^{-1}\Big|\varphi(a_{l_q+1})|_{J_q}\Big|\cdots\Big]\nonumber\\ &\quad\quad\in \bigoplus_{J}C_{k+q}(\underline{\mathscr{C}}(U_{I\cup J})). \end{align} $$

where the sum is over all ordered $J=(j_1,...,j_q)$ disjoint from I and all $0\leq l_1\leq l_2\leq ...\leq l_q\leq k$ . The sign is given by

$$ \begin{align*}\epsilon=(|a_0|+...+|a_{l_1}|+l_1)+...+(|a_0|+...+|a_{l_q}|+l_q)\end{align*} $$

where $\sigma (I,J)$ is the sign of the permutation which arranges $(j_1,...,j_q,i_0,...,i_p)$ in order. For example if

are three composable arrows in $\underline {\mathscr {C}}(U_I)$ and $j\notin I$ then the terms in $h^1(a_0[a_1|a_2])$ over $U_{I\cup j}$ are in bijection with paths of length 4 in the diagram

We define ${\check{C}}(\varphi ,\alpha ):{\check{C}}(\mathscr {U},\underline {C}^{?}_{\bullet }(\underline {\mathscr {C}}))\to {\check{C}}(\mathscr {U},\underline {C}^{?}_{\bullet }(\underline {\mathscr {C}}))$ by

$$ \begin{align*}C_k(\underline{\mathscr{C}}(U_I))\ni \bar{a}:=a_0[a_1|\cdots|a_k]\mapsto \sum_{q\geq 0}h^q(\bar{a})\end{align*} $$

and extending over infinite sums in the case $?=II$ .

Proposition 11. The map ${\check{C}}(\varphi ,\alpha )$ is a chain map and fits into a commutative diagram in $D_{\mathbb {Z}/2}(\mathbb {C})$

where the vertical arrows are induced by the restriction functors.

Proof. We first check that ${\check{C}}(\varphi ,\alpha )$ is a chain map. This follows from the following lemma

Lemma 12. (i) $d_{{\check{C}}}\circ h^{q-1}+\bar {d}_{2}\circ h^{q}=h^{q}\circ \bar {d}_{2}+h^{q-1}\circ d_{{\check{C}}}$ for $q\geq 1$ , (ii) $\bar {d}_1\circ h^q=h^q\circ \bar {d}_1$ for all $q\geq 0$ , (iii) $\bar {d}_0\circ h^q=h^q\circ \bar {d}_0$ for all $q\geq 0$ .

Proof of lemma.

For $(i)$ we note that $\bar {d}_{2}\circ h^q(\bar {a})$ will consist of terms of the following form

$$ \begin{align*}&(I) \ \alpha\varphi(a_0|_{U_{I\cup J}})\alpha^{-1}\Big[\cdots \Big]\\ &(II) \ \alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots \Big|\varphi((a_ra_{r+1})|_{U_{J_s}})|_{U_{I\cup J}}\Big|\cdots\Big]\\ &(III) \ \alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots \Big|\varphi(a_{l_s}|_{U_{J_{q-s+1}}})|_{U_{I\cup J}}\alpha^{-1}|_{U_{I\cup J}}\Big|\cdots\Big]\\ &(IV) \ \alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots \Big|\alpha^{-1}|_{U_{I\cup J}}\varphi(a_{l_{s}+1}|_{U_{J_{q-s}}})|_{U_{I\cup J}}\Big|\cdots\Big]\\ &(V) \ \alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots\Big|\alpha^{-1}|_{U_{I\cup J}}\alpha^{-1}|_{U_{I\cup J}}\Big|\cdots \Big]\\ &(VI) \ \alpha^{-1}|_{U_{I\cup J}}\alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots \Big]\\ &(VII) \ \varphi(a_k)|_{U_{I\cup J}}\alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots\Big] \end{align*} $$

The terms in $(I)$ cancel with the terms in $d_{{\check{C}}}\circ h^{q-1}(\bar {a})$ , the terms in $(II)$ cancel with terms in $h^q\circ \bar {d}_2(\bar {a})$ , the terms in $(III)$ cancel with the terms in $(IV)$ , the terms in $(V)$ cancel with themselves, the terms in $(VI)$ cancel with the terms from $h^{q-1}\circ d_{{\check{C}}}$ and finally the terms $(VII)$ cancel with the remaining terms in $h^q\circ \bar {d}_2(\bar {a})$ .

Part $(ii)$ and $(iii)$ is also just a matter of checking.

Now for the second part of the proposition we note that the two composites are related by a homotopy H which we now describe. On an element $\bar {a}=a_0[a_1|\cdots |a_k]$ it is defined to be

$$ \begin{align*}\sum_{q\geq 1}\tilde{h}^q(\bar{a})\end{align*} $$

where $\tilde {h}^q:C_k(\underline {\mathscr {C}}(X))\to {\check{C}}^{q-1}(\mathscr {U},\underline {C}_{k+q})$ are defined exactly like $h^q$ in (4) with the convention that $p=-1$ .

We say that two lax-morphisms $(\varphi ,\alpha ),(\psi ,\beta ):\underline {\mathscr {C}}\to \underline {\mathscr {D}}$ are isomorphic if there are natural isomorphisms $\tau _U:\varphi _U\overset {\sim }{\implies } \psi _U$ such that for any $V\subset U$ we have a commutative diagram of natural isomorphisms

Proof. Let $\tau :(\varphi ,\alpha )\to (\psi ,\beta )$ be an isomorphism. The following is a chain homotopy between ${\check{C}}(\varphi ,\alpha )$ and ${\check{C}}(\psi ,\beta )$ :

$$ \begin{align*} &\sum_{q\geq 0}H^q,\text{ where }H^q(a_0[a_1|\cdots|a_k]) :=\\ \sum\sum &(-1)^{\epsilon'}\tau|_{U_{I\cup J}}\alpha\varphi(a_0|_{U_{I\cup J}})\Big[\cdots\Big|\alpha^{-1}\Big|\varphi(a_{l_i+1}|_{U_{J_{q-i}}})|_{U_{I\cup J}}\Big|\cdots\\ &\Big|\tau|_{U_{I\cup J}}^{-1}\Big|\cdots\Big|\beta^{-1}|_{U_{I\cup J}}\Big|\psi(a_{l_{i+1}+1})|_{U_{I\cup J}}\Big|\cdots\Big] \end{align*} $$

where the outer sum is indexed as in the definition of $h^q$ (see equation $(4)$ above) and the inner sum runs from $0$ to $k+q$ picking out the position for $\tau ^{-1}$ . The sign is

$$ \begin{align*}\epsilon'=\epsilon+|a_0|+...+|a_r|+r+s\end{align*} $$

where $\epsilon $ is as in the definition of $h^q$ , $a_r$ is the last morphism among $a_0,...,a_k$ that appear before $\tau ^{-1}$ and s is the number of $\alpha ^{-1}$ ’s that appear before $\tau ^{-1}$ .

If $\underline {\mathscr {C}}$ is a presheaf of dg algebras (recall that this means $\underline {\mathscr {C}}(U)$ only has one object for all $U\subset X$ ), then any two lax morphisms of the form $(\varphi ,\alpha ),(\varphi ,\beta ):\underline {\mathscr {C}}\to \underline {\mathscr {D}}$ are isomorphic. Indeed, lets call the object in $\underline {\mathscr {C}}(X)\, c$ . Then we can define $\tau _{U,c|_U}:=(\beta ^X_U)^{-1}\circ \alpha ^X_U$ . Now if $\varphi $ is an actual morphism of presheaves of dg categories then $(\varphi ,\mathrm {id})$ is a lax morphism and if $\underline {\mathscr {C}}$ is a presheaf of dg algebras then any other lax morphism $(\varphi ,\alpha )$ is isomorphic to $(\varphi ,\mathrm {id})$ . Note that a morphism of presheaves of dg categories $\varphi $ induces a morphism

$$ \begin{align*}{\check{C}}(\varphi):{\check{C}}(\mathscr{U},\underline{C}_{\bullet}(\underline{\mathscr{C}}))\to {\check{C}}(\mathscr{U},\underline{C}_{\bullet}(\underline{\mathscr{D}}))\end{align*} $$

by taking $(\varphi _{U_I})_{\ast }$ on the summand $C_{\bullet }(\underline {\mathscr {C}}(U_I))$ .

Proof. Note that with $\alpha $ being the identity natural transformation all the $h^q$ ’s in the definition of ${\check{C}}(\varphi ,\mathrm {id})$ , with $q>1$ , vanish. Therefore ${\check{C}}(\varphi ,\mathrm {id})-{\check{C}}(\varphi )=h^1$ . Let $\bar {a}=a_0[a_1|\cdots |a_k]\in C_k(\underline {\mathscr {C}}(U_I))$ , where $I=\{i_0<...<i_p\}$ and define

$$ \begin{align*}H(\bar{a})=\sum_{j,l_1,l_2}(-1)^{\tau}\varphi (a_0)[\cdots|\varphi(a_{l_1})|\mathrm{id}|\cdots|\varphi(a_{l_2})|\mathrm{id}|\cdots]\in \bigoplus C_{k+2}(\underline{\mathscr{D}}(U_{I\cup j}))\end{align*} $$

where the sum is over all $j\notin I$ , and $0\leq l_1\leq l_2\leq k$ and the sign is given by

$$ \begin{align*}\tau=(|a_0|+...+|a_{l_1}|+l_1)+(|a_0|+...+|a_{l_1}|+l_2)+\sigma(I,j).\end{align*} $$

Then $(\bar {d}_1+\bar {d}_2+d_{{\check{C}}})H+H(\bar {d}_1+\bar {d}_2+d_{{\check{C}}})=-h^1={\check{C}}(\varphi )-{\check{C}}(\varphi ,\mathrm {id})$ .

3.1 Affine matrix factorizations

Let A be a commutative ring and $h\in A$ and consider the cdg algebra $A_h=(A,0,h)$ from Example 3. Then a matrix factorization of h is nothing but an object in $A_h-mod^{cdg}_{fgp}$ . More precisely, such an object consists of the following data

where $P^0,P^1$ are projective A-modules, $\delta ^0,\delta ^1$ are A-module homomorphisms such that the composites $\delta ^1\delta ^0$ and $\delta ^0\delta ^1$ are both multiplication by h.

3.2 Global matrix factorizations

Now let X be a smooth quasi-compact scheme and $f\in \mathcal {O}_X(X)$ a global function. We will define the derived category of coherent matrix factorizations and a dg enhancement of this. First, consider the dg category $\text {fact}_{nv}(X,f)$ whose objects consist of the following data

where $\mathcal {F}^0,\mathcal {F}^1$ are coherent $\mathcal {O}_X$ -modules and $\delta ^0,\delta ^1$ are $\mathcal {O}_X$ -module homomorphisms such that $\delta ^0\delta ^1 $ and $\delta ^1\delta ^0$ both equal multiplication by f. The hom spaces are defined as the usual ‘naive’ hom spaces between complexes:

$$ \begin{align*} \text{Hom}_{\text{fact}_{nv}(X,f)}(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet}):=\text{Hom}_{\mathcal{O}_X}^{\bullet}(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet}) \end{align*} $$

with differential $\partial (\alpha ):=\delta _{\mathcal {G}^{\bullet }}\alpha -(-1)^{|\alpha |}\alpha \delta _{\mathcal {F^{\bullet }}}$ . The category $Z^0(\text {fact}_{nv}(X,f))$ is abelian and $H^0(\text {fact}_{nv}(X,f))$ is triangulated. We define $D(\text {fact}_{nv}(X,f))$ as the Verdier quotient of $H^0(\text {fact}_{nv}(X,f))$ by the full thick triangulated subcategory generated by all totalizations of short exact sequences in $Z^0(\text {fact}_{nv}(X,f))$ .

Next we define $\text {vect}_{nv}(X,f)$ to be the full dg subcategory of $\text {fact}_{nv}(X,f)$ on the objects whose underlying sheaves $\mathcal {F}^0,\mathcal {F}^1$ are vector bundles. We define as well $D(\text {vect}_{nv}(X,f))$ to be the Verdier quotient of $H^0(\text {vect}_{nv}(X,f))$ by the full thick triangulated subcategory generated by all totalizations of short exact sequences in $Z^0(\text {vect}_{nv}(X,f))$ . There is a canonical functor $D(\text {vect}_{nv}(X,f))\to D(\text {fact}_{nv}(X,f))$ which is an equivalence by [Reference Ballard, Favero and Katzarkov1] proposition 3.14.

If $X=\text {Spec}(A)$ is affine then this is the category $mod^{cdg}_{fgp}-A_f$ defined earlier and if X is smooth and affine then $H^0(\text {vect}_{nv}(X,f))\simeq D(\text {fact}_{nv}(X,f))$ . Under some extra assumptions on f this gives an important example where the two kinds of Hochschild homology agree.

If X is not affine the category $\text {vect}_{nv}(X,f)$ will not work as a dg enhancement for $D(\text {fact}_{nv}(X,f))$ . Instead we have to introduce the dg category $\text {vect}_{{\check{C}}}(X,f)$ . Let $\mathscr {U}$ denote a finite affine cover of X. Then we define $\text {vect}_{{\check{C}}}(X,f)$ to have objects same as $\text {vect}_{nv}(X,f)$ but for the hom complexes we take

(5) $$ \begin{align} \text{Hom}_{\text{vect}_{{\check{C}}}(X,f)}(\mathcal{P}^{\bullet},\mathcal{Q}^{\bullet}):=Tot({\check{C}}{^{\bullet}}(\mathscr{U},\underline{\text{Hom}}^{\bullet}(\mathcal{P}^{\bullet},\mathcal{Q}^{\bullet}))). \end{align} $$

Composition is defined using the Alexander-Whitney product from Section 1.2. This is a dg enhancement of $D(\text {fact}_{nv}(X,f))$ when X is smooth (see [Reference Lin and Pomerleano11] Proposition 2.11).

We will also need the cdg category $\text {qfact}_{nv}(X,f)$ . Its objects are defined exactly as the objects of $\text {fact}_{nv}(X,f)$ except we don’t require anything about the composites $\delta ^0\delta ^1$ or $\delta ^1\delta ^0$ . The hom-spaces and the differentials are defined in the same way as for $\text {fact}_{nv}(X,f)$ and the curvature of an object $\mathcal {F}^{\bullet }$ is the endomorphism $\delta ^2-\rho _f$ . We define $\text {qvect}_{nv}(X,f)$ to be the full sub cdg category of $\text {qfact}_{nv}$ whose underlying sheaves are vector bundles. We define the cdg category $\text {qvect}_{{\check{C}}}(X,f)$ by altering the hom complexes as in (5).

Finally we will also need presheaf versions of some of these. We define the following presheaf of dg categories

$$ \begin{align*} \underline{\text{vect}}_{{\check{C}}}(X,f)(U):=\text{vect}_{{\check{C}}}(U,f|_U). \end{align*} $$

The presheaves of cdg categories $\underline {\text {qvect}}_{{\check{C}}}(X,f)$ , $\underline {\text {vect}_{nv}}(X,f)$ and $\underline {\text {qvect}_{nv}}(X,f)$ are defined similarly.

3.3 Hochschild homology of $\text {vect}_{{\check{C}}}(X,f)$

The Hochschild homology of $\text {vect}_{{\check{C}}}(X,f)$ will be denoted $HH_{\bullet }(X,f)$ . For a smooth variety X where $f|_{X\setminus V(f)}$ is smooth this was computed in [Reference Efimov7], it is derived global sections of the complex of sheaves $(\Omega ^{\bullet }_X,-df\wedge (-))$ . Below is a slight adaption of the computation in [Reference Efimov7].

There is a sequence of maps which we will show are all quasi-isomorphisms.

(6)

We will first discuss $(6)$ (see [Reference Căldăraru and Tu4], Theorem 4.2(b)).

Proposition 17. The following defines a quasi-isomorphism

$$ \begin{align*} &HKR_{(X,f)}:{\check{C}}(\mathscr{U},\underline{C}_{\bullet}^{II}(\mathcal{O}_f))\to {\check{C}}(\mathscr{U},(\Omega_X^{\bullet},df\wedge-)),\\ C_{\bullet}^{II}(\mathcal{O}_f(U_{i_0\cdots i_p}))&\ni a_0[a_1|\cdots |a_k]\mapsto \frac{1}{k!}a_0da_1\wedge...\wedge da_k\in \Omega^k_X(V\cap U_{i_0\cdots i_p}) \end{align*} $$

To prove this we will need two lemmas.

Lemma 18. Suppose that we have an unbounded exact chain complex of inverse systems of abelian groups

$$ \begin{align*} S_{\bullet}:=(...\to S_n\to S_{n-1}\to S_{n-2}\to...). \end{align*} $$

and that each term is Mittag-Leffler. Then the complex

$$ \begin{align*} ...\to\lim S_n\to \lim S_{n-1}\to\lim S_{n-2}\to... \end{align*} $$

is exact.

Proof of lemma.

Split the long exact sequence into short exact sequences

Now the inverse systems $B_k=Z_{k-1}$ are quotients of Mittag-Leffler systems and are therefore themselves Mittag-Leffler. It follows that after applying $\underset {\leftarrow }{\lim }$ termwise the diagonal short exact sequences remain exact and then it follows that the middle long exact sequence remains exact.

Proof. Consider the inverse systems of chain complexes

$$ \begin{align*}S(A)&:=\{...\to\text{Tot}^{\oplus}A_{\geq -2,\bullet}\to \text{Tot}^{\oplus} A_{\geq -1,\bullet}\to \text{Tot}^{\oplus}A_{\geq 0,\bullet}\}\\ S(B)&:=\{...\to \text{Tot}^{\oplus}B_{\geq -2,\bullet}\to \text{Tot}^{\oplus} B_{\geq -1,\bullet}\to \text{Tot}^{\oplus}B_{\geq 0,\bullet}\} \end{align*} $$

where the maps are induced by canonical quotient maps of bicomplexes. The map $\alpha $ induces a morphism $S(\alpha ):S(A)\to S(B)$ of complexes of inverse systems. The assumption on $\alpha $ and the Mapping Theorem for filtered modules ([Reference Mac Lane12], Theorem XI.3.4) imply that $S(\alpha )$ is a termwise quasi-isomorphism, in other words a quasi-isomorphism of complexes of inverse systems. But then $Cone(S(\alpha ))$ is an acyclic complex of inverse systems of abelian groups, all of whose terms are Mittag-Leffler and therefore, by the previous lemma,

$$ \begin{align*} Cone(\underset{\leftarrow}{\lim}S(\alpha))=\underset{\leftarrow}{\lim}Cone(S(\alpha))=0. \end{align*} $$

We see that $\underset {\leftarrow }{\lim }S(\alpha ):\text {Tot}^{\prod }A_{\bullet ,\bullet }\to \text {Tot}^{\prod }B_{\bullet ,\bullet }$ is a quasi-isomorphism.

Proof of proposition.

The affine case follows from the classical HKR theorem and by applying the previous lemma to the bicomplexes

and

For the non-affine case, think of $\text {Tot}^{\prod }A_{\bullet ,\bullet }$ and $\text {Tot}^{\prod }B_{\bullet ,\bullet }$ as presheaves of complexes. We want to show that

$$ \begin{align*} {{\check{C}}}{^p}(\mathscr{U},(\text{Tot}^{\prod}A_{\bullet,\bullet})_q)\to{{\check{C}}}{^p}(\mathscr{U},(\text{Tot}^{\prod}B_{\bullet,\bullet})_q) \end{align*} $$

is a quasi-isomorphism. By considering total complexes and filtering this in the p-direction we obtain a map of filtered complexes. It then follows from the Mapping Theorem of filtered complexes and the affine case already considered that this map of bicomplexes gives rise to a quasi-isomorphism of total complexes. This proves the proposition.

Now $(5)$ is a quasi-isomorphism because for any $U_I=U_{i_0\cdots i_p}$ the map $C^{II}_{\bullet }(\mathcal {O}_{U_I,-f|_{U_I}})\to C^{II}_{\bullet }(\text {qvect}_{nv}(U_I,f|_{U_I}))$ is a quasi-isomorphism by Proposition 7. The map $(4)$ is a quasi-isomorphism for exactly the same reasons. The map $(3)$ is a quasi-isomorphism because each $C_{\bullet }(\text {vect}_{nv}(U_I,f|_{U_I}))\to C^{II}_{\bullet }(\text {vect}_{nv}(U_I,f|_{U_I}))$ is a quasi-isomorphism by Proposition 15. The map $(2)$ is a quasi-isomorphism because for each $U_I$ we have a quasi-equivalence $\text {vect}_{nv}(U_I,f|_{U_I})\to \text {vect}_{{\check{C}}}(U_I,f|_{U_I})$ which induces quasi-isomorphisms on standard complexes. The proof that the map $(1)$ is a quasi-isomorphism is similar to the proof of Proposition 5.1 in [Reference Efimov7]. We include the proof here with some adjustments to make it fit our situation.

Proof. We prove it for $\mathscr {C}=\underline {\text {vect}}_{{\check{C}}}(X,f)$ . Recall that $\mathscr {U}$ is referring to a fixed affine open covering $X=U_1\cup ...\cup U_n$ . Let $U=U_n$ and $V=U_1\cup ...\cup U_{n-1}$ . Let $Z=X\setminus U$ . For any open $W\subset X$ let $\mathscr {C}(W)_{Z\cap W}\subset \mathscr {C}(W)$ denote the full sub dg category on the objects which become acyclic in $\mathscr {C}(W\cap U).$ Let $\mathscr {V}$ denote the open covering $V=U_1\cup U_3\cup ...\cup U_{n-1}$ . Let $\mathscr {C}'$ denote the presheaf of dg categories on V

$$ \begin{align*}V\supset W\mapsto \text{vect}_{{\check{C}}(\mathscr{V})}(W,f|_W).\end{align*} $$

We have a natural functor $\mathscr {R}es: \mathscr {C}(X)\to \mathscr {C}'(V)$ which restricts objects on X to V and which is defined on homomorphism complexes by

$$ \begin{align*}{\check{C}}(\mathscr{U},\underline{\text{Hom}}_X^{\bullet}(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet}))\to {\check{C}}(\mathscr{U}\cap V,\underline{\text{Hom}}_V^{\bullet}(\mathcal{F}|_V^{\bullet},\mathcal{G}|_V^{\bullet}))\twoheadrightarrow {\check{C}}(\mathscr{V},\underline{\text{Hom}}_V^{\bullet}(\mathcal{F}|_V^{\bullet},\mathcal{G}|_V^{\bullet})).\end{align*} $$

After passing to homotopy categories this functor fits into a commutative diagram

We have the following diagram in $D_{Z/2}(\mathbb {C})$

where the rows are distinguished. (See [Reference Efimov7], Proposition 5.1 and its proof.) Since $Z\subset V$ the leftmost and the rightmost vertical arrows are quasi-isomorphisms. Therefore we get a Mayer-Viertoris exact triangle

We now claim that there is an exact triangle of the form

to which the first exact triangle naturally maps. Assuming this claim for now the rest of the proof goes as follows. Because the open covering $\mathscr {U}\cap U$ of U has U itself as one of its open sets the map $C_{\bullet }(\mathscr {C}(U))\to {\check{C}}(\mathscr {U}\cap U,\underline {C}_{\bullet }(\mathscr {C}))$ is a quasi-isomorphism. By induction on the number of open sets the maps $C_{\bullet }(\mathscr {C}'(V))\to {\check{C}}(\mathscr {V},\underline {C}_{\bullet }(\mathscr {C}'))$ and $C_{\bullet }(\mathscr {C}'(V\cap U))\to {\check{C}}(\mathscr {V}\cap U,\underline {C}_{\bullet }(\mathscr {C}'))$ are quasi-isomorphisms. The proposition then follows by the five lemma.

To construct the second distinguished triangle, we first define what the two maps $\alpha $ and $\beta $ are. First, let $A\subset {\check{C}}(\mathscr {U},\underline {C}_{\bullet }(\mathscr {C}))$ be the subcomplex consisting of all summands $C_{\bullet }(\mathscr {C}(U_I))$ with $n\in I$ and $B=C_{\bullet }(\mathscr {C}(U_I))/A$ so that as a vector space we have $C_{\bullet }(\mathscr {C}(U_I))=A\oplus B$ . Similarly let $E\subset {\check{C}}(\mathscr {U}\cap U,\underline {C}_{\bullet }(\mathscr {C}))$ be the sub complex consisting of the summands $C_{\bullet }(\mathscr {C}(U_I\cap U_n))$ with $n\in I$ and $F={\check{C}}(\mathscr {U}\cap U,\underline {C}_{\bullet }(\mathscr {C}))/E$ and $G={\check{C}}(\mathscr {V},\underline {C}_{\bullet }(\mathscr {C}'))$ so that as vector spaces we have ${\check{C}}(\mathscr {U}\cap U,\underline {C}_{\bullet }(\mathscr {C}))\oplus {\check{C}}(\mathscr {V},\underline {C}_{\bullet }(\mathscr {C}'))=E\oplus F\oplus G$ . Finally let $H={\check{C}}(\mathscr {V}\cap U,\underline {C}_{\bullet }(\mathscr {C}'))$ . In this notation, the maps $\alpha $ and $\beta $ can be described by matrices

$$ \begin{align*}\alpha=\begin{bmatrix}\mathrm{id} & \ 0 & \ 0\\0& \ \alpha_0 & \alpha_1 \end{bmatrix}:A\oplus B\to E\oplus F\oplus G\end{align*} $$

$$ \begin{align*}\beta=\begin{bmatrix}0\\ \beta_0\\\beta_1\end{bmatrix}:E\oplus F\oplus G\to H\end{align*} $$

where $\alpha _0$ is made up by the restrictions $C_{\bullet }(\mathscr {C}(U_I))\to C_{\bullet }(\mathscr {C}(U_I\cap U))$ ( $n\notin I$ ), $\alpha _1$ is a quasi-isomorphism which is induced by the quasi-equivalences $\mathscr {C}(U_I)\to \mathscr {C}'(U_I)$ ( $n\notin I$ ). The map $\beta _0$ is also a quasi-isomorphism, it’s induced by the quasi-equivalences $\mathscr {C}(U_I\cap U)\to \mathscr {C}'(U_I\cap U)$ ( $n\notin I$ ) and $\beta _1$ is $-1$ times the map induced by the restrictions $\mathscr {C}'(U_I)\to \mathscr {C}'(U_I\cap U)$ ( $n\notin I$ ). In this situation one can show that the canonical map $Cone(\alpha )\to H$ is a quasi-isomorphism and therefore we obtain the required distinguished triangle.

3.4 The case $f=0$

When $f=0$ we have two interesting categories.

On the one hand the $\mathbb {Z}$ -graded dg-category $\text {perf}_{{\check{C}}}(X)$ whose objects are finite $\mathbb {Z}$ -graded complexes of vector bundles and whose morphism complexes are defined as

$$ \begin{align*}\text{Hom}_{\text{perf}_{{\check{C}}}(X)}(\mathcal{P}^{\bullet},\mathcal{Q}^{\bullet}):=Tot({\check{C}}^{\bullet}(\mathscr{U},\underline{\text{Hom}}_X^{\bullet}(\mathcal{P}^{\bullet},\mathcal{Q}^{\bullet}))).\end{align*} $$

We can then think of $\text {perf}_{{\check{C}}}(X)$ as a $\mathbb {Z}/2$ -graded category by forgetting the $\mathbb {Z}$ -grading on the homomorphism complexes and only remembering the parity of each degree.

On the other hand we can consider the $\mathbb {Z}/2$ -graded category $\text {vect}_{{\check{C}}}(X,0)$ defined in Section 3.2. Note that we have a fully faithful dg functor $\text {perf}_{{\check{C}}}(X)\to \text {vect}_{{\check{C}}}(X,0)$ which forgets the $\mathbb {Z}$ -grading on the objects and only remembers the parity of each term of a complex. In fact, we have the following proposition.

Proof. Let . Let $\mathcal {K}^i=\ker (\delta ^i)$ and $\mathcal {Q}^i=\text {im }(\delta ^i)$ . Then the following is an exact triangle in $D(\text {fact}_{nv}(X,0))$

Both $(\mathcal {K}^{\bullet },0)$ and $(\mathcal {Q}^{\bullet },0)$ are isomorphic to objects in $H^0(\text {perf}_{{\check{C}}}(X))$ . The reason is because X is non-singular the components of $\mathcal {K}^{\bullet }$ and $\mathcal {Q}^{\bullet }$ have finite resolutions by locally free sheaves. It follows that $\mathcal {F}^{\bullet }$ too is isomorphic to an object in $H^0(\text {perf}_{{\check{C}}}(X))$ .

Now let’s consider the sequence

From each term in the above sequence there is a natural map to the corresponding term of (6) forming commutative squares, the map ${\check{C}}(\mathscr {U},\Omega _X^{\bullet })\to {\check{C}}(\mathscr {U},\Omega _X^{\bullet })$ being the identity. All of the maps above are quasi-isomorphisms and this forces all of the maps from this sequence to (6) to be quasi-isomorphisms

Finally we record here also the following fact.

Lemma 22. Let X be a smooth variety. Let $\mathcal {F}\in \text {vect}_{nv}(X,0)$ (which just means it’s a $\mathbb {Z}/2$ graded complex of finite rank vector bundles). There is a natural isomorphism

$$ \begin{align*}\text{Hom}_{D(\text{vect}_{nv}(X,0))}(\mathcal{O}_X,\mathcal{F})\overset{\sim}{\to}H^0({\check{C}}^{\bullet}(\mathscr{U},\mathcal{F})).\end{align*} $$

Proof. First, we claim that there is an injective chain map $\mathcal {F}\to \mathcal {I}$ where $\mathcal {I}$ is a $\mathbb {Z}/2$ -graded complex of injective sheaves on X whose cone is absolutely acyclic. Indeed this is completely dual to the argument in the proof of proposition 3.24 of [Reference Ballard, Favero and Katzarkov1]. This gives the first map below. The second isomorphism follows from lemma 2.22 in [Reference Ballard, Deliu, Favero, Isik and Katzarkov2]. We get

$$ \begin{align*}\text{Hom}_{D(\text{vect}_{nv}(X,0))}(\mathcal{O}_X,\mathcal{F})&\overset{\sim}{\to} \text{Hom}_{D(\text{fact}_{nv}(X,0))}(\mathcal{O}_X,\mathcal{I})\\ &\overset{\sim}{\leftarrow} H^0(\text{Hom}_{\text{fact}_{nv}(X,0)}(\mathcal{O}_X,\mathcal{I}))=H^0(\Gamma(X,\mathcal{I})).\end{align*} $$

But we also have $H^0(\Gamma (X,\mathcal {I}^i))\overset {\sim }{\to }H^0({\check{C}}(\mathscr {U},\mathcal {I}^i))$ for $i=0,1$ since $\mathcal {I}^i$ are injective and because the covering is finite a spectral sequence argument gives a quasi-isomorphism of $\mathbb {Z}/2$ -graded complexes $\Gamma (X,\mathcal {I})\overset {\sim }{\to } {\check{C}}(\mathscr {U},\mathcal {I})$ . Finally over each affine open V we have a quasi-isomorphism $\Gamma (V,\mathcal {F})\overset {\sim }{\to } \Gamma (V,\mathcal {I})$ and these assemble to a quasi-isomorphism ${\check{C}}(\mathscr {U},\mathcal {F})\overset {\sim }{\to } {\check{C}}(\mathscr {U},\mathcal {I})$ . The desired map is the zig-zag of isomorphisms

4.1 Setup

Let X be a smooth complex variety and let us fix a finite affine covering $\mathscr {U}$ of X. Let $f\in H^0(X,\mathcal {O}_X)$ and suppose $Y\subset X$ is a smooth divisor such that $f|_Y=0$ . We assume also that $f|_{X\setminus f^{-1}(0)}$ is smooth. The categories we are mostly interested in are $D^b(Y):=D^b(\text {coh}Y)$ (usual derived category) and $D(\text {fact}_{nv}(X,f))$ .

We have dg enhancements for both $D^b(Y)$ and $D(\text {fact}_{nv}(X,f))$ . For $D^b(Y)$ we take the category $\text {perf}_{{\check{C}}}(Y)$ described in Section 3.4 and for $D(\text {fact}_{nv}(X,f))$ we take $\text {vect}_{{\check{C}}}(X,f)$ described in Section 3.2. We have a functor $i_{\ast }:D^b(Y)\to D(\text {fact}_{nv}(X,f))$ .

We now explain how $i_{\ast }$ lifts to a quasi-functor $\underline {i}_{\ast }:\text {perf}_{ {\check{C}}}(Y)\dashrightarrow \text {vect}_{ {\check{C}}}(X,f)$ . Recall that by quasi-functor we mean a dg functor $\underline {i}_{\ast }:\text {perf}_{ {\check{C}}}(Y)\to \text {perf}(\text {vect}_{ {\check{C}}}(X,f))$ (see Section 2.4). To define it we need to introduce the dg category $\text {fact}_{{\check{C}}}(X,f)$ which has the same objects as $\text {fact}_{nv}(X,f)$ but the homomorphism complexes are computed as in $\text {vect}_{{\check{C}}}(X,f)$ . This dg category is not the dg enhancement of any interesting triangulated category and we only introduce it to simplify notation in what follows. Now the dg functor $\underline {i}_{\ast }$ is defined on $\mathcal {F}\in \text {perf}_{{\check{C}}}(Y)$ by

$$ \begin{align*}\underline{i}_{\ast}(\mathcal{F})(-):=\text{Hom}_{\text{fact}_{{\check{C}}}(X,f)}(-,i_{\ast}\mathcal{F}):\text{vect}_{{\check{C}}}(X,f)\to Com(\mathbb{C}).\end{align*} $$

To see that this is perfect we use that there is a resolution, by which we mean a degree zero cycle in $\text {fact}_{nv}(X,f)$ whose cone is absolutely acyclic, $\mathcal {V}\overset {\sim }{\to }i_{\ast }\mathcal {F}$ from an object in $\mathcal {V}\in \text {vect}_{{\check{C}}}(X,f)$ (see for example proposition 3.14 of [Reference Ballard, Favero and Katzarkov1] and its proof). This induces a morphism

$$ \begin{align*}\text{Hom}_{\text{vect}_{{\check{C}}}}(-,\mathcal{V})\to \text{Hom}_{\text{fact}_{{\check{C}}}(X,f)}(-,i_{\ast}\mathcal{F})\end{align*} $$

which is a quasi-isomorphism because we only plug in objects of $\text {vect}_{{\check{C}}}(X,f)$ into the first slot.

Now recall that $\text {fact}_{nv}(X,f)$ , $\text {vect}_{nv}(X,f)$ denote the ‘naive’ dg categories whose homomorphism complexes are just the usual $Hom$ -complexes between complexes (see Section 3.2). Similarly, we will denote by $\text {perf}_{nv}(Y)$ the dg category whose objects are finite complexes of vector bundles and whose homomorphism complexes are the ‘naive’ hom-complexes. Recall also from Section 3.2 that $D(\text {fact}_{nv}(X,f))$ and $D(\text {vect}_{nv}(X,f))$ are the triangulated categories obtained as Verdier quotients of $H^0(\text {fact}_{nv}(X,f))$ and $H^0(\text {vect}_{nv}(X,f))$ by the subcategories of absolutely acyclic objects. Similarly, we define $D(\text {perf}_{nv}(Y))$ to be the Verdier quotient of $H^0(\text {perf}_{nv}(Y))$ by the subcategory of acyclic complexes and note that the canonical functor $D(\text {perf}_{nv}(Y))\to D^b(Y)$ is an equivalence.

In the following lemma we explain in what sense the quasi-functor $\underline {i}_{\ast }$ lifts the exact functor $i_{\ast }$ .

Lemma 23. There is a commutative (up to isomorphism) diagram of exact functors

where the arrows labelled $\sim $ are equivalences. Moreover, $Yoneda$ and the functor $Y'$ induce isomorphisms in Hochschild homology.

Proof. Before dealing with commutativity we need to explain what the vertical arrows are.

The leftmost vertical arrow comes from identifying $D^b(cohY)$ with the full subcategory on the perfect complexes, $D(perf_{nv}(Y))\subset D^b(cohY)$ . Now, $D(\text {perf}_{nv}(Y))=H^0(\text {perf}_{nv}(Y))/Acyc(\text {perf}_{nv}(Y))$ . The obvious dg functor $\text {perf}_{nv}(Y)\to \text {perf}_{{\check{C}}}(Y)$ induces an exact functor $H^0(\text {perf}_{nv}(Y))\to H^0(\text {perf}_{{\check{C}}}(Y))$ and this descends to the Verdier quotient $D(\text {perf}_{nv}(Y))\to H^0(\text {perf}_{{\check{C}}}(Y))$ .

To define the middle vertical arrow we first observe that there is a dg functor $\text {fact}_{nv}(X,f)\to \text {perf}(\text {vect}_{{\check{C}}}(X,f))$ defined by $\mathcal {H}\mapsto \text {Hom}_{\text {fact}_{{\check{C}}}(X,f)}(-,\mathcal {H})$ . This induces an exact functor $H^0(\text {fact}_{nv}(X,f))\to H^0(\text {perf}(\text {vect}_{{\check{C}}}(X,f)))$ which descends to a functor out of the Verdier quotient $D(\text {fact}_{nv}(X,f))\to H^0(\text {perf}(\text {vect}_{{\check{C}}}(X,f)))$ .

For the right vertical map, we note that there is a dg functor $\text {vect}_{nv}(X,f)\to \text {vect}_{{\check{C}}}(X,f)$ . This gives a map $H^0(\text {vect}_{nv}(X,f))\to H^0(\text {vect}_{{\check{C}}}(X,f))$ which descends to a functor out of the verdier quotient $D(\text {vect}_{nv}(X,f))\to H^0(\text {vect}_{{\check{C}}}(X,f))$ .

Now the left square commutes on the nose if we start with something in $D(\text {perf}_{nv}(Y))$ .

The rightmost square too commutes on the nose.

Next, the fact that the leftmost vertical functor is an equivalence is well known. That the bottom right horizontal functor is an equivalence is for example in Proposition 3.14 in [Reference Ballard, Favero and Katzarkov1]. That the right vertical arrow is an equivalence follows from the fact that the map on hom spaces factors as two isomorphisms

where the first isomorphism comes from the adjunction in Corollary 3.28 of [Reference Ballard, Favero and Katzarkov1] and the second isomorphism is Lemma 22.

Finally, the fact that $Yoneda$ induces an isomorphism in Hochschild homology has been discussed in Section 2.4 and then $Y'$ is forced to induce an isomorphism in Hochschild homology by the commutativity of the right square in the diagram of the lemma.

Now the quasi-functor $\underline {i}_{\ast }$ induces a map in Hochschild homology as described in Section 2.4. We will compute this map in terms of the identifications

$$ \begin{align*}HH_{\bullet}(Y):=HH_{\bullet}(\text{perf}_{{\check{C}}}(Y))\simeq {\check{C}}(\mathscr{U}\cap Y,\Omega_Y^{\bullet})\end{align*} $$

and

$$ \begin{align*}HH_{\bullet}(X,f):=HH_{\bullet}(\text{vect}_{{\check{C}}}(X,f))\simeq {\check{C}}(\mathscr{U},(\Omega^{\bullet}_X,-df\wedge(-)))\end{align*} $$

from Sections 3.4 and 3.3 respectively.

4.2 Replacing $\underline {i}_{\ast }$ by a dg functor

Consider the following matrix factorization

where $x\in H^0(X,\mathcal {O}_X(Y))$ is an equation for Y and $g\in H^0(X,\mathcal {O}_X(-Y))$ is such that $gx=f\in H^0(X,\mathcal {O}_X)$ . In $D(\text {fact}_{nv}(X,f))$ we have $i_{\ast }\mathcal {O}_Y\cong P$ . Indeed, the kernel of the canonical map $P\to i_{\ast }\mathcal {O}_Y$ is the matrix factorization

which is zero in the derived category because it maps to zero under the equivalence $\text {Cok}:D(\text {fact}_{nv}(X,f))\to D^b_{sing}(V(f))$ (see [Reference Orlov14] for details about this equivalence).

Consider also the sheaf of dg algebras $\mathcal {A}:=[\mathcal {O}_X(-Y)\cdot e\overset {x}{\to } \mathcal {O}_X]$ where e is a formal variable that squares to zero. It is quasi-isomorphic to $\mathcal {O}_Y$ as sheaves of dg algebras on X. The quasi-isomorphism $\mathcal {A}\to \mathcal {O}_Y$ also induces a quasi-isomorphism on Hochschild complexes for any affine open $V\subset X$ , $C_{\bullet }(\mathcal {A}|_V)\overset {\sim }{\to }C_{\bullet }(\mathcal {O}_Y|_V)$ .

We define the dg category $\text {perf}_{{\check{C}}}(\mathcal {A})$ to have objects, same as $\text {perf}_{{\check{C}}}(X)$ . The homomorphism complexes are given by

$$ \begin{align*} \text{Hom}_{\text{perf}_{{\check{C}}}(\mathcal{A})}(\mathcal{V},\mathcal{W}):={\check{C}}(\mathscr{U},\underline{\text{Hom}}_{\mathcal{A}}^{\bullet}(\mathcal{V}\otimes \mathcal{A},\mathcal{W}\otimes \mathcal{A})). \end{align*} $$

We now introduce a dg functor $\text {Ind}:\text {perf}_{{\check{C}}}(\mathcal {A})\to \text {perf}_{{\check{C}}}(Y)$ . On objects it is defined by

$$ \begin{align*}\text{Ind}(\mathcal{V})=i^*\mathcal{V}.\end{align*} $$

The chain maps on homomorphism complexes are induced by the following morphism of complexes of sheaves

Lemma 24. The functor Ind defined above induces an isomorphism in $D_{Z/2}(\mathbb {C})$

$$ \begin{align*}Ind_{\ast}:C_{\bullet}(\text{perf}_{{\check{C}}}(\mathcal{A}))\to C_{\bullet} (perf_{{\check{C}}}(Y)).\end{align*} $$

Proof. For a dg category $\mathscr {C}$ , let $\tilde {\mathscr {C}}$ denote its pretriangulated hull. We get a strictly commutative diagram of dg functors

where the vertical functors induce isomorphisms in Hochschild homology ([Reference Keller9]) and the right vertical arrow is a quasi-equivalence since $\text {perf}_{{\check{C}}}(Y)$ is pretriangulated. Therefore it is enough to check that $\widetilde {\text {Ind}}$ induces an isomorphism in Hochschild homology. We show that $\widetilde {\text {Ind}}$ is in fact a quasi-equivalence. To see that $\widetilde {\text {Ind}}$ induces a quasi-isomorphism on homomorphism complexes we note that $\text {Ind}$ applied to hom-complexes can be factored as a composite of quasi-isomorphisms:

$$ \begin{align*} \text{Hom}_{\text{perf}_{{\check{C}}}(\mathcal{A})}(\mathcal{V}^{\bullet}\otimes \mathcal{A},\mathcal{W}^{\bullet}\otimes \mathcal{A})&= {\check{C}}(\mathscr{U},\underline{\text{Hom}}^{\bullet}_{\mathcal{A}}(\mathcal{V}^{\bullet}\otimes \mathcal{A},\mathcal{W}^{\bullet}\otimes \mathcal{A}))\\ &\simeq {\check{C}}(\mathscr{U},\underline{\text{Hom}}^{\bullet}_{\mathcal{O}_X}(\mathcal{V}^{\bullet},\mathcal{W}^{\bullet}\otimes \mathcal{A}))\\ &\simeq {\check{C}}(\mathscr{U},(\mathcal{V}^\vee)^{\bullet}\otimes \mathcal{W}^{\bullet}\otimes \mathcal{A})\\ &\overset{\sim}{\to} {\check{C}}(\mathscr{U},(\mathcal{V}^\vee)^{\bullet}\otimes \mathcal{W}^{\bullet}\otimes \mathcal{O}_Y)\\ & \simeq {\check{C}}(\mathscr{U},\underline{\text{Hom}}^{\bullet}_{\mathcal{O}_X}(\mathcal{V}^{\bullet},\mathcal{W}^{\bullet}\otimes \mathcal{O}_Y))\\ &\simeq {\check{C}}(\mathscr{U},\underline{\text{Hom}}^{\bullet}_{\mathcal{O}_Y}(\mathcal{V}^{\bullet}\otimes \mathcal{O}_Y,\mathcal{W}^{\bullet}\otimes \mathcal{O}_Y))\\ &\simeq \text{Hom}_{\text{perf}_{{\check{C}}}(Y)}(\text{Ind}(\mathcal{V}^{\bullet}\otimes \mathcal{A}),\text{Ind}(\mathcal{W}^{\bullet}\otimes \mathcal{A})) \end{align*} $$

This means that $H^0(\text {Ind})$ is fully faithful and therefore so is $H^0(\widetilde {\text {Ind}})$ . Therefore, to show that $H^0(\widetilde {\text {Ind}})$ is essentially surjective it is enough to check that the smallest triangulated subcategory of $H^0(\widetilde {\text {perf}_{{\check{C}}}(Y)})$ containing the image of $H^0(\widetilde {\text {Ind}})$ is all of $H^0(\widetilde {\text {perf}_{{\check{C}}}(Y)})$ . Let $\mathcal {O}_X(1)$ denote a very ample line bundle on X and let $\mathcal {O}_Y(1)=\mathcal {O}_X(1)|_Y$ . We know that $\{\mathcal {O}_Y(n)\}_{n\in \mathbb {Z}}$ classically generates $H^0(\text {perf}_{{\check{C}}}(Y))\simeq \widetilde {H^0(\text {perf}_{{\check{C}}}(Y)})$ . Essential surjectivity then follows from the observation that

$$ \begin{align*}\mathcal{O}_Y(n)=\text{Ind}( \mathcal{O}_X(n)).\end{align*} $$

Let $\mathcal {A}^\#$ denote the underlying sheaf of graded algebras of $\mathcal {A}$ and $P^\#$ denote the underlying sheaf of $\mathbb {Z}/2$ -graded $\mathcal {O}_X$ -modules of P. As graded $\mathcal {O}_X$ -modules we have $\mathcal {A}^\#=P^\# $ and this gives us a multiplication $\mu :\mathcal {A}^\#\otimes P^\#\to P^\#$ . If $\mathcal {M}$ is an $\mathcal {A}$ -module we can thus make sense of the tensor product $\mathcal {M}\otimes _{\mathcal {A}} P$ as a graded $\mathcal {O}_X$ -module. In fact, the morphism

$$ \begin{align*} \mathcal{M}\otimes \mathcal{A}\otimes P\to \mathcal{M}\otimes P, \ m\otimes a\otimes p\mapsto ma\otimes p-m\otimes ap \end{align*} $$

whose cokernel by definition is the tensor product $\mathcal {M}\otimes _{\mathcal {A}} P$ is a morphism of matrix factorizations of f. Therefore the product $\mathcal {M}\otimes _{\mathcal {A}} P$ is naturally a matrix factorization of f. We get a dg functor

(7) $$ \begin{align} -\otimes_{\mathcal{A}} P:\text{perf}_{{\check{C}}}(\mathcal{A})\to \text{vect}_{{\check{C}}}(X,f) \end{align} $$

which on objects is defined by $\mathcal {V}\mapsto \mathcal {V}\otimes _X P$ and the chain maps on hom complexes are induced by the morphism of complexes of sheaves

Recall that $\underline {i}_{\ast }:\text {perf}_{{\check{C}}}(Y)\to \text {perf}(\text {vect}_{{\check{C}}}(X,f))$ was defined in Section 4.1.

Lemma 25. The following diagram commutes up to natural quasi-isomorphism

Proof. For any $Q\in \text {vect}_{{\check{C}}}(X,f)$ , the natural map

$$ \begin{align*}\text{Hom}_{\text{vect}_{{\check{C}}}(X,f)}(Q,\mathcal{V}\otimes P)\to \text{Hom}_{\text{fact}_{{\check{C}}}(X,f) }(Q,\mathcal{V}\otimes \mathcal{O}_Y)\end{align*} $$

is a quasi-isomorphism where we recall that $\text {fact}_{{\check{C}}}(X,f)$ is the DG category with objects same as $\text {fact}_{nv}(X,f)$ and whose homomorphism complexes are defined in the same way as in $\text {vect}_{{\check{C}}}(X,f)$ .

In fact, we can define similar functors

$$ \begin{align*}\text{Ind}_U&:\text{perf}_{{\check{C}}}(\mathcal{A}|_U)\to \text{perf}_{{\check{C}}}(Y\cap U),\\ -\otimes_{\mathcal{A}|_U}P|_U&:\text{perf}_{{\check{C}}}(\mathcal{A}|_U)\to \text{vect}_{{\check{C}}}(U,f|_U) \end{align*} $$

for any open set U and assemble them into lax morphisms of presheaves of dg categories.

Proposition 26. There are natural isomorphisms

$$ \begin{align*}\alpha_{U,V,\mathcal{F}}: \text{Ind}_V(\mathcal{F}|_V)\implies \text{Ind}_U(\mathcal{F})|_V\end{align*} $$

and

$$ \begin{align*}\beta_{U,V,\mathcal{F}}:\mathcal{F}|_V\otimes_{\mathcal{A}|_V}P|_V\implies (\mathcal{F}\otimes_{\mathcal{A}|_U}P|_U)|_V\end{align*} $$

making

$$ \begin{align*}(\text{Ind},\alpha):\underline{\text{perf}}_{{\check{C}}}(\mathcal{A})\to \underline{\text{perf}}_{{\check{C}}}(Y)\end{align*} $$

and

$$ \begin{align*}(-\otimes_{\mathcal{A}} P,\beta):\underline{\text{perf}}_{{\check{C}}}(\mathcal{A})\to \underline{\text{vect}}_{{\check{C}}}(X,f)\end{align*} $$

into lax morphisms of presheaves of dg categories.

Proof. For $Ind$ this is just the statement that for $W\subset U\subset V$ we have a commutative diagram of natural isomorphisms

It may be easier to see that the following diagram commutes

where $j_W:W\to U$ and $j_V:V\to U$ are the inclusions.

The case of $\otimes _{\mathcal {A}} P$ is similar.

4.3 HKR-type map for $\mathcal {A}$

Let $\Omega ^k_X(\log Y)$ denote the algebraic k-forms on X with logarithmic poles along Y. It is a subsheaf of $\Omega _X^k(Y)$ whose sections locally look like $\omega +\frac {dx}{x}\wedge \omega '$ where $\omega \in \Omega _X^k$ , $\omega '\in \Omega ^{k-1}_X$ and x is a local equation for Y. There is a short exact sequence of complexes

(8)

where the first map is the inclusion and the second map is locally defined by $\beta +\frac {dx}{x}\wedge \alpha \mapsto \alpha |_Y$ . It gives rise to a quasi-isomorphism

(9) $$ \begin{align} \Omega_{\mathcal{A}}^{\bullet}:=\text{Cone}(L)\overset{\sim}{\to}\Omega^{\bullet}_Y. \end{align} $$

We will construct a commutative diagram

(10)

The horizontal arrows are induced by the quasi-isomorphisms $\mathcal {A}\to \mathcal {O}_Y$ and (9) respectively and the left vertical arrow comes from the classical HKR theorem. It remains to define the map $HKR_{\mathcal {A}}$ . Let $x_i\in \mathcal {O}_X(U_i)$ be local equations for $Y\subset X$ . Let $u_{ij}=x_jx_i^{-1}$ on the intersections $U_{ij}$ . Then we identify $\mathcal {A}|_{U_j}\cong [\mathcal {O}_{U_i}\epsilon _i\to \mathcal {O}_{U_i}]$ where $\epsilon _i$ is in degree $-1$ and the differential maps $\epsilon _i$ to $x_i$ . Note that on the intersections we have $u_{ij}\epsilon _i=\epsilon _j$ . Let $U_I=U_{i_0\cdots i_p}$ and let $\bar {a}:=a_0[a_1|\cdots |a_k]\in {\mathcal {A}^0(U_I)}^{\otimes k+1}\subset C_k(\mathcal {A}(U_I))$ . Then define

$$ \begin{align*} HKR_{\mathcal{A}}(\bar{a})&:=\frac{1}{k!}a_0 \frac{dx_{i_0}}{x_{i_0}}\wedge da_1\wedge...\wedge da_k\\ &+\frac{1}{k!}a_0dx_{i_0}\wedge dg_{i_0}\wedge da_1\wedge...\wedge da_k\\ &-\sum_{j<i_0}\frac{(-1)^p}{k!}a_0\frac{du_{i_0j}}{u_{i_0j}}\wedge da_1\wedge...\wedge da_k\\ &\in \Gamma\Big(U_I,\Omega^{k+1}_X(\log Y))\oplus \Gamma(U_I,\Omega_X^{k+2})\oplus (\bigoplus_{j<i_0}\Gamma(U_{j\cup I},\Omega_X^{k+1})\Big) \end{align*} $$

and

$$ \begin{align*} HKR_{\mathcal{A}}(a_0[a_1|\cdots|a_l\epsilon_{i_0}|\cdots|a_k]):= \frac{(-1)^{l}}{k!}a_0dx_{i_0}\wedge da_1\wedge ...\wedge da_k. \end{align*} $$

and $HKR_{\mathcal {A}}$ vanishes on elements which have two or more factors from $\mathcal {A}^{-1}$ .

Proof. We check that $HKR_{\mathcal {A}}$ is a chain map at the level of presheaves. We do this first on elements of internal degree zero. As before let $U_I=U_{i_0\cdots i_p}$ and let $\bar {a}:=a_0[a_1|\cdots |a_k]\in \mathcal {A}^0(U_I)^{\otimes k+1}\subset C_k(\mathcal {A}(U_I))$ .

(11) $$ \begin{align} HKR_{\mathcal{A}}(\bar{d}_2\bar{a})&=0\nonumber\\ HKR_{\mathcal{A}}(\bar{d}_1\bar{a})&=0\nonumber\\ HKR_{\mathcal{A}}(d_{Cech}\bar{a})&=HKR_{\mathcal{A}}\Big(\sum_{j\notin I}(-1)^{Sgn(j,I)}\bar{a}|_{U_{I\cup j}}\Big)\nonumber\\ &=HKR_{\mathcal{A}}\Big(\sum_{j<i_0}\bar{a}|_{U_{I\cup j}}+\sum_{j\notin I, \ j>i_0}(-1)^{Sgn(j,I)}\bar{a}|_{U_{I\cup j}}\Big)\nonumber\\ &=\frac{1}{k!}\sum_{j<i_0}a_0\frac{dx_{j}}{x{j}}\wedge da_1\wedge...\wedge da_k|_{j\cup I}\nonumber \\ &+\frac{1}{k!}\sum_{j<i_0}a_0dx_{j}\wedge dg_{j}\wedge da_1\wedge...\wedge da_k|_{j\cup I}\nonumber\\ &-\frac{1}{k!}\sum_{j'<j<i_0}(-1)^{p+1}a_0\frac{du_{jj'}}{u_{jj'}}\wedge da_1\wedge ...\wedge da_k|_{j'\cup j\cup I}\nonumber\\ &+HKR_{\mathcal{A}} \Big(\sum_{j\notin I, \ j>i_0}(-1)^{Sgn(j,I)}\bar{a}|_{U_{I\cup j}}\Big). \end{align} $$

On the other hand we have

(12) $$ \begin{align} d_{Cech}HKR_{\mathcal{A}}(\bar{a})&=\frac{1}{k!}\sum_{j<i_0}a_0 \frac{dx_{i_0}}{x_{i_0}}\wedge da_1\wedge...\wedge da_k|_{j\cup I} \nonumber \\ &+\frac{1}{k!}\sum_{j<i_0}a_0 dx_{i_0}\wedge dg_{i_0}\wedge da_1\wedge...\wedge da_k|_{j\cup I} \nonumber \\ &-\frac{1}{k!}\sum_{j<j'<i_0}(-1)^pa_0\Big(\frac{du_{i_0j'}}{u_{i_0j'}}-\frac{du_{i_0j}}{u_{i_0j}}\Big)\wedge da_1\wedge...\wedge da_k|_{j\cup j'\cup I} \nonumber \\ &+\sum_{j\notin I,j>i_0}HKR_{\mathcal{A}}(\bar{a})|_{{j\cup I}} \end{align} $$

and

(13) $$ \begin{align} \bar{d}_{Cone}HKR_{\mathcal{A}}(\bar{a})&=\frac{(-1)^{p}}{k!}a_0df\wedge \frac{dx_{i_0}}{x_{i_0}}\wedge da_1\wedge...\wedge da_k\nonumber \\ &+\frac{(-1)^{p}}{k!}L(a_0 dx_{i_0}\wedge dg_{i_0}\wedge da_1\wedge...\wedge da_k) \nonumber \\ &+\frac{1}{k!}\sum_{j<i_0}a_0df\wedge \frac{du_{i_0j}}{u_{i_0j}}\wedge da_1\wedge...\wedge da_k|_{j\cup I}\nonumber \\ &+\frac{1}{k!}\sum_{j<i_0}L(a_0 \frac{du_{i_0j}}{u_{i_0j}}\wedge da_1\wedge...\wedge da_k)|_{j\cup I} \end{align} $$

From here one can check that $(\bar {d}_{Cone}+d_{Cech})HKR_{\mathcal {A}}(\bar {a})=HKR_{\mathcal {A}}(d_{Cech}+\bar {d}_1+\bar {d}_2)(\bar {a})$ . Indeed the first sum in (11) cancels with the first sum in (12) and the last sum in (13). The second sum in (11) cancels with the second sum in (12) and the sum on the third row of (13). The third sum in (11) cancels with the third sum in (12). The last sum in (11) cancels with the last sum in (12). Finally the first two terms in (13) cancel with each other.

Next, let’s check that $(\bar {d}_{Cone}+d_{Cech})HKR_{\mathcal {A}}(\bar {b})=HKR_{\mathcal {A}}(d_{Cech}+\bar {d}_1+\bar {d}_2)(\bar {b})$ where $\bar {b}=b_0[b_1|\cdots |b_l\epsilon _{i_0}|\cdots |b_k]$ has internal degree $-1$ . We have

$$ \begin{align*} HKR_{\mathcal{A}}(\bar{d}_2\bar{b})&=0\\ HKR_{\mathcal{A}}(\bar{d}_1\bar{b})&=(-1)^{l+p}\frac{1}{k!}b_0\frac{dx_{i_0}}{x_{i_0}}\wedge db_1\wedge...\wedge d(b_lx_{i_0})\wedge...\wedge db_k\\ &+(-1)^{l+p}\frac{1}{k!}b_0dx_{i_0}\wedge dg_{i_0}\wedge db_1\wedge...\wedge d(b_lx_{i_0})\wedge...\wedge db_k\\ &-\sum_{j<i_0}(-1)^{l}\frac{1}{k!} b_0\frac{du_{i_0j}}{u_{i_0j}}\wedge db_1\wedge...\wedge d(b_lx_{i_0})\wedge...\wedge db_k\\ HKR_{\mathcal{A}}(d_{Cech}\bar{b})&=\sum_{j\notin I, \ j>i_0}(-1)^{l+Sgn(j,I)}\frac{1}{k!}b_0dx_{i_0}\wedge db_1\wedge...\wedge db_l\wedge...\wedge db_k\\ &+\sum_{j<i_0}(-1)^{l}\frac{1}{k!}b_0dx_{j}\wedge db_1\wedge...\wedge d(u_{ji_0}b_l)\wedge...\wedge db_k \end{align*} $$

On the other hand we have

$$ \begin{align*} d_{Cech}HKR_{\mathcal{A}}(\bar{a})&=\sum_{j\notin I}(-1)^{Sgn(j,I)+l}\frac{1}{k!}b_0dx_{i_0}\wedge db_1\wedge...\wedge db_l\wedge...\wedge db_k\\ \bar{d}_{Cone}HKR_{\mathcal{A}}(\bar{b})&=(-1)^{l+p+1}\frac{1}{k!}b_0df\wedge dx_{i_0}\wedge db_1\wedge...\wedge db_l\wedge...\wedge db_k\\ &+(-1)^{l+p}\frac{1}{k!}L(b_0dx_{i_0}\wedge db_1\wedge...\wedge db_l\wedge...\wedge db_k) \end{align*} $$

Again the relevant equality can be checked from these formulas. This proves the first part of the proposition. The last thing one has to check for the first part is that $HKR_{\mathcal {A}}\circ \bar {d}_1$ vanishes on elements with two factors from $\mathcal {A}^{-1}$ .

The second part can be checked from the definitions. The last part then follows from the second because all the other arrows in the diagram are quasi-isomorphisms.

4.4 Multiplying by the Todd class

We have the wedge product map $\Omega ^{\bullet }_X\otimes \Omega _X^{\bullet }\to \Omega _X^{\bullet }$ which we can use to make ${\check{C}}(\mathscr {U},\Omega _X^{\bullet })$ into a dg algebra (see Section ?? for the precise formula). The natural map ${\check{C}}(\mathscr {U},\Omega _X^{\bullet })\to {\check{C}}(\mathscr {U},\Omega _Y^{\bullet })$ is a dg algebra homomorphism. We make ${\check{C}}(\mathscr {U},\Omega _{\mathcal {A}}^{\bullet })$ into a left dg ${\check{C}}(\mathscr {U},\Omega _X^{\bullet })-$ module as follows. For $\alpha \in {\check{C}}^p(\mathscr {U},\Omega _X^q)$ , $\beta \in {\check{C}}^a(\mathscr {U},\Omega _X^b)$ and $\gamma \in {\check{C}}^{a'}(\mathscr {U},\Omega _X^{b'}(\log Y))$ we define

$$ \begin{align*}(\alpha\cdot \beta)_{i_0\cdots i_{p+a}}=(-1)^{qa}\alpha_{i_0\cdots i_p}|_{U_{i_0\cdots i_{p+a}}}\wedge \beta_{i_p\cdots i_{p+a}}|_{U_{i_0\cdots i_{p+a}}}\end{align*} $$

and

$$ \begin{align*}(\alpha\cdot \gamma)_{i_0\cdots i_{p+a'}}=(-1)^{q(a'+1)}\alpha_{i_0\cdots i_p}|_{U_{i_0\cdots i_{p+a'}}}\wedge \gamma_{i_p\cdots i_{p+a'}}|_{U_{i_0\cdots i_{p+a'}}}.\end{align*} $$

The Cech cocycle $(u_{ij})_{j<i}\in {\check{C}}{^1}(\mathscr {U},\mathcal {O}_X^\times )$ corresponds to the line bundle $\mathcal {O}_X(-Y)$ . The Chern class $c_1(-Y)$ is the Cech cocycle $(u_{ij}^{-1}{du_{ij}})_{j<i}\in {\check{C}}{^1}(\mathscr {U},\Omega ^1_X)$ . The relative Todd class of $Y\subset X$ , denoted $Td_{Y/X}$ is defined as the formal power series $\frac {x}{1-e^{-x}}$ evaluated at $c_1(-Y)$ . Its inverse is

$$ \begin{align*} \text{Td}_{Y/X}^{-1}:=-\sum_{q\geq 0}\frac{c_1(-Y)^{\wedge q}}{(q+1)!}. \end{align*} $$

Proposition 28. The map

$$ \begin{align*} Td_{Y/X}^{-1}\wedge (-):{\check{C}}(\mathscr{U},\Omega_{\mathcal{A}}^{\bullet})\to {\check{C}}(\mathscr{U},\Omega^{\bullet}_{\mathcal{A}}) \end{align*} $$

commutes with the differentials and gives rise to a commutative diagram

4.5 Trace map for a small cdg category of quasi-modules

Let $\{P_1,...,P_r\}$ be a collection of objects in $\text {qvect}_{nv}(X,f)$ and let $\mathscr {C}$ be the presheaf of cdg categories defined by $U\mapsto \langle P_1|_U,...,P_r|_U\rangle $ where the latter refers to the full sub cdg category of $\text {qvect}_{nv}(U,f|_U)$ . Assume that we have trivializations $P_i|_{U_j}\cong \mathcal {O}_{U_j}^{n_i}\oplus \mathcal {O}_{U_j}^{m_i}$ . Our goal is to define a map

(14) $$ \begin{align} \phi:{\check{C}}(\mathscr{U},\underline{C}^{II}_{\bullet}(\mathscr{C}))\to {\check{C}}(\mathscr{U},\underline{C}^{II}_{\bullet}(\mathcal{O}_{-f}))\end{align} $$

such that if $(P_i,\delta _{P_i})=(\mathcal {O}_X,0)$ for some $1\leq i\leq r$ then $\phi $ is a quasi-inverse to the quasi-isomorphism

(15) $$ \begin{align}{\check{C}}(\mathscr{U},\underline{C}_{\bullet}^{II}(\mathcal{O}_{-f}))\to {\check{C}}(\mathscr{U},\underline{C}^{II}_{\bullet}(\mathscr{C}))\end{align} $$

induced by the Yoneda functor.

Given a morphism $\alpha :P_i(U_l)\to P_j(U_l)$ we will denote by $(\alpha )_l$ the matrix defined by

$$ \begin{align*}\mathcal{O}_{U_l}^{n_i}\oplus \mathcal{O}_{U_l}^{m_i}\cong P_i\rightarrow P_j\cong \mathcal{O}_{U_l}^{n_j}\oplus \mathcal{O}_{U_l}^{m_j}.\end{align*} $$

Let $g_{kl}$ denote the change of basis matrix such that

$$ \begin{align*}g_{kl}(\alpha)_lg_{kl}^{-1}=(\alpha)_k\end{align*} $$

on $U_k\cap U_l$ .

We will also need the preasheaf of cdg categories $\mathscr {F}$ which to an open set U assigns the subcategory of $\text {qvect}_{nv}(U,f|_U)$ on all objects of the form

We will define a collection of maps

$$ \begin{align*}h^q:{\check{C}}(\mathscr{U},\underline{C}_{\bullet}^{II}(\mathscr{C}))\to {\check{C}}(\mathscr{U},\underline{C}_{\bullet}^{II}(\mathscr{F})). \end{align*} $$

Let $\bar {\alpha }:=\alpha _0[\alpha _1|\cdots |\alpha _k]\in C_{\bullet }^{II}(\mathscr {C})(U_{i_0\cdots i_p})$ and define

$$ \begin{align*}h^q(\bar{\alpha}) & :=\\ & \sum(-1)^{\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[(\alpha_1)_{i_{-q}}\Big|\cdots\Big|(\alpha_{l_1})_{i_{-q}}\Big|g_{i_{1-q}i_{-q}}^{-1}\Big|(\alpha_{l_1+1})_{i_{1-q}}\Big|\cdots \\ & \cdots\Big|(\alpha_{l_2})_{i_{1-q}}\Big|g_{i_{2-q}i_{1-q}}^{-1}\Big|(\alpha_{l_2+1})_{i_{2-q}}\Big|\cdots \Big|(\alpha_{l_q})_{i_{-1}}\Big|g_{i_{0}i_{-1}}^{-1}\Big|(\alpha_{l_q+1})_{i_{0}}\Big|\cdots\Big] \end{align*} $$

where the sum is over all q-tuples $1\leq i_{-q}<i_{1-q}<i_{2-q}<...<i_{-1}<i_{0}$ and all q-tuples $0\leq l_1\leq ...\leq l_q\leq k$ . The sign is given by

$$ \begin{align*}\epsilon=\sum_{i=1}^q\Big(\sum_{j=0}^{l_i}|\alpha_j|+l_i\Big) \end{align*} $$

Lemma 29. We have

$$ \begin{align*} \bar{d}_2h^q+d_{Cech} h^{q-1}=h^{q-1}d_{Cech}+h^q\bar{d_2}. \end{align*} $$

Proof. We apply both sides to $\bar {\alpha }=\alpha _0[\alpha _1|\cdots |\alpha _k]$ which is a section over $U_{i_0...i_p}$ . First, $\bar {d}_2h^q(\bar {\alpha })$ will consist of terms of the form

(16) $$ \begin{align} &\color{black}(-1)^{p+q+|\alpha_0|+\epsilon+pq+\binom{q}{2}}g_{i_0i_{1-q}}(\alpha_0)_{i_{1-q}}\Big[(\alpha_1)_{i_{1-q}}\Big|\cdots \Big], \nonumber \\ &\color{black}(-1)^{p+q+|\alpha_0|+...+|\alpha_{l_j}|+l_j+j+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots\Big|g_{i_{j+1-q}i_{j-1-q}}^{-1}\Big|\cdots\Big],\nonumber \\ &\color{black}(-1)^{p+q+|\alpha_0|+...+|\alpha_{r}|+r+j+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots \Big|(\alpha_{r}\alpha_{r+1})_{j-q}\Big|\cdots\Big], \nonumber \\ &\color{black}(-1)^{p+q+j+1+|\alpha_0|+|\alpha_1|+...+|\alpha_{l_{j}}|+l_j+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots \Big|g^{-1}_{i_{j-q}i_{j-1-q}}(\alpha_{l_{j}+1})_{i_{j-q}}\Big|\cdots\Big] \nonumber \\ &\color{black}(-1)^{p+q+j+|\alpha_0|+...+|\alpha_{l_{j}+1}|+l_{j}+1+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots \Big|(\alpha_{l_{j}+1})_{i_{j-1-q}}g^{-1}_{i_{j-q}i_{j-1-q}}\Big|\cdots\Big] \nonumber \\ &\color{black}(-1)^{p+q+1+|\alpha_0|+...+|\alpha_k|+k+q-1+\epsilon+pq+\binom{q}{2}}g_{i_{-1}i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots \Big], \nonumber \\ &\color{black}(-1)^{p+q+1+(|\alpha_k|+1)(|\alpha_0|+...+|\alpha_{k-1}|+k-1+q)+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_k\alpha_0)_{i_{-q}}\Big[\cdots \Big] \end{align} $$

Next, $d_{Cech}h^{q-1}$ will consist of terms of the form

(17) $$ \begin{align} \begin{aligned} &\color{black}(-1)^{\epsilon+p(q-1)+\binom{q-1}{2}}g_{i_0i_{1-q}}(\alpha_0)_{i_{1-q}}\Big[(\alpha_1)_{i_{1-q}}\Big|\cdots\Big],\\ &\color{black}(-1)^{\epsilon+p(q-1)+\binom{q-1}{2}+j}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots\Big|g_{i_{j+1-q}i_{j-1-q}}^{-1}\Big|\cdots\Big],\\ &\color{black}(-1)^{\epsilon+p(q-1)+\binom{q-1}{2}+q-1+l+1}h^{q-1}(\bar{\alpha})|_{i_{-q}i_{1-q}...i_0...i_{l}j...i_p} \end{aligned} \end{align} $$

Also, $h^{q-1}d_{Cech}$ will consist of terms of the form

(18) $$ \begin{align} \begin{aligned} &\color{black}(-1)^{l+1+\epsilon+(p+1)(q-1)+\binom{q-1}{2}}h^{q-1}(\bar{\alpha}|_{i_0...i_lj...i_p})\\ &\color{black}(-1)^{\epsilon+(p+1)(q-1)+\binom{q-1}{2}}g_{i_{-1}i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots \Big]. \end{aligned} \end{align} $$

Finally, $h^q\bar {d}_2$ will contain terms of the form

(19) $$ \begin{align}\begin{aligned} &\color{black}(-1)^{p+|\alpha_0|+...+|\alpha_r|+r+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_0)_{i_{-q}}\Big[\cdots \Big|(\alpha_{r}\alpha_{r+1})_{j-q}\Big|\cdots\Big]\\ &\color{black}(-1)^{p+1+(|\alpha_k|+1)(|\alpha_0|+|\alpha_1|+...+|\alpha_{k-1}|+k-1)+\epsilon+pq+\binom{q}{2}}g_{i_0i_{-q}}(\alpha_k\alpha_0)_{i_{-q}}\Big[\cdots \Big] \end{aligned}\end{align} $$

Here the $\epsilon '$ in the signs are all different. The first term in (16) will cancel with the first term in (17). The second term in (16) cancels with the second term in (17). The third term in (16) cancels with the first term in (19). The fourth and fifth terms in (16) cancel. The sixth term in (16) cancels with the second term in (18). The last term in (16) cancels with the last term in (19). Finally the last term in (17) cancels with the first term in (18).

Now we introduce the second ingredient to the map (14). Suppose R is a commutative ring and $P_0,...,P_m$ are a collection of $\mathbb {Z}/2$ -graded free modules. Pick homogeneous bases $\{e_i^j\}$ for each $P_i$ . Let $F^0\in \text {Hom}(P_1,P_0)$ , $F^1\in \text {Hom}(P_2,P_1)$ ,..., $F^m\in \text {Hom}(P_0,P_m)$ be endomorphisms which we can think of as matrices with the fixed choice of bases. We define

$$ \begin{align*} sTr(F^0\otimes F^1\otimes...\otimes F^m):=\sum(-1)^{\sigma}F^0_{j_0j_1}\otimes F^1_{j_1j_2}\otimes...\otimes F^m_{j_mj_0} \end{align*} $$

where the sum is over all $j_0,...,j_m$ and the sign is given by

$$ \begin{align*} \sigma=(m+1)|e_0^{j_0}|+|e_1^{j_1}|+...+|e_m^{j_m}|. \end{align*} $$

Extending this over infinite sums gives a map of presheaves $\underline {C}^{II}_{\bullet }(\mathscr {F})\to \underline {C}^{II}_{\bullet }(\mathcal {O}_{-f})$ from which we obtain a map ${\check{C}}(\mathscr {U},\underline {C}^{II}_{\bullet }(\mathscr {F}))\to {\check{C}}(\mathscr {U},\underline {C}^{II}_{\bullet }(\mathcal {O}_f))$ .

We have to introduce one more bit of notation before defining the map $\phi $ . For $\bar {\alpha }:=\alpha _0[\alpha _1|\cdots |\alpha _k]\in C^{II}_{\bullet }(\mathscr {C}(V))$ let

$$ \begin{align*} Sh(\underset{n factors}{\underbrace{[\delta|\cdots|\delta]}},\bar{\alpha})=\sum_{i_0+...+i_k=n} \alpha_0[\delta^{i_0}|\alpha_1|\delta^{i_1}| \alpha_2|\cdots|\alpha_k|\delta^{i_k} ]. \end{align*} $$

For each $n\geq 0$ , this defines a map of presheaves $\underline {C}^{II}_{\bullet }(\mathscr {C})\to \underline {C}^{II}_{\bullet }(\mathscr {C})$ and from it we obtain a map ${\check{C}}(\mathscr {U},\underline {C}^{II}_{\bullet }(\mathscr {C}))\to {\check{C}}(\mathscr {U,}\underline {C}^{II}_{\bullet }(\mathscr {C}))$ .

Definition 30. We define

$$ \begin{align*}\phi:{\check{C}}(\mathscr{U},\underline{C}^{II}_{\bullet}(\mathscr{C}))\to{\check{C}}(\mathscr{U},\underline{C}^{II}_{\bullet}(\mathcal{O}_{-f})) \end{align*} $$

by

$$ \begin{align*} \phi=\sum_{n,q\geq 0}(-1)^nsTr\Big(h^q\big(Sh(\underset{n \ factors}{\underbrace{[\delta|\cdots|\delta]}},-)\big)\Big) \end{align*} $$

Proof. First note that $\bar {d_0}+\bar {d_2}+d_{Cech}$ commutes with $sTr$ . It suffices to check that

$$ \begin{align*}\sum (-1)^nh^{q,n}(d_{Cech}+\bar{d_0}+\bar{d_1}+\bar{d_2})=(d_{Cech}+\bar{d_0}+\bar{d_2})\sum (-1)^nh^{q,n}.\end{align*} $$

Comparing Cech-Hochschild bidegree this equality comes down to

$$ \begin{align*} \begin{cases} (1) \ \ \ (-1)^{n}h^{q,n}\circ \bar{d_1}+\\ (2) \ \ \ (-1)^{n+1}h^{q-1,n+1}\circ d_{Cech}+\\ (3) \ \ \ (-1)^{n-1}h^{q,n-1}\circ \bar{d_0}+\\ (4) \ \ \ (-1)^{n+1}h^{q,n+1}\circ \bar{d_2} \end{cases}=\begin{cases} (A) \ \ \ (-1)^{n+1}d_{Cech}\circ h^{q-1,n+1}+\\ (B) \ \ \ (-1)^{n-1}\bar{d_0}\circ h^{q,n-1}+\\ (C) \ \ \ (-1)^{n+1}\bar{d_2}\circ h^{q,n+1} \end{cases} \end{align*} $$

Then using Lemma 29

$$ \begin{align*} (A)+(C)& =(-1)^{n+1}d_{Cech}h^{q-1,n+1}(\bar{\alpha})+(-1)^{n+1}\bar{d_2}h^{q,n+1}(\bar{\alpha}) \\ &=(-1)^{n+1}d_{Cech}h^{q-1,0}(Sh(\delta^{n+1},\bar{\alpha}))+(-1)^{n+1}\bar{d_2}h^{q,0}(Sh(\delta^{n+1},\bar{\alpha}))\\ &=(-1)^{n+1}h^{q-1,0}d_{Cech}(Sh(\delta^{n+1},\bar{\alpha}))+(-1)^{n+1}h^{q,0}\bar{d_2}Sh(\delta^{n+1},\bar{\alpha})\\ &=(-1)^{n+1}h^{q-1,n+1}d_{Cech}(\bar{\alpha})+(-1)^nh^{q,n}\bar{d_1}\bar{\alpha}\\ &\quad +(-1)^{n+1+p}h^{q,n-1}Sh(\delta^2,\bar{\alpha})+(-1)^{n+1}h^{q,n+1}\bar{d_2}(\bar{\alpha})\\ &=(2)+(1)+(4)+(3)-(B). \end{align*} $$

One can compute that the composite $\underline {{\check{C}}}(\mathscr {U},\underline {C}_{\bullet }^{II}(\mathcal {O}_{-f}))\to \underline {{\check{C}}}(\mathscr {U},\underline {C}^{II}_{\bullet }(\mathscr {C}))\to \underline {{\check{C}}}(\mathscr {U},\underline {C}_{\bullet }^{II}(\mathcal {O}_{-f}))\to \underline {{\check{C}}}(\mathscr {U},(\Omega _X^{\bullet },-df\wedge -))$ equals the map $HKR_{(X,-f)}$ from Proposition 17. From this the second part follows.

4.6 Main theorem

Recall that $Y\subset X$ is locally cut out by $x_i\in \mathcal {O}_X(U_i)$ and that $u_{ij}x_i=x_j$ over $U_{ij}$ . Recall also the matrix factorization P from Section 4.2. Over each of the open sets $U_i$ we can identify $P=\mathcal {O}_X\epsilon _i\oplus \mathcal {O}_X$ . Suppose $F_i$ is a matrix describing the action of an element of $\mathrm {End}(P|_{U_i})$ in this basis. If $F_j$ denotes the matrix of the same endomorphism but with respect to the basis $P=\mathcal {O}_X\epsilon _j\oplus \mathcal {O}_X$ over $U_{ji}$ then $g_{ij}F_jg_{ij}^{-1}=F_i$ where the change of basis matrix is

$$ \begin{align*} g_{ij}= \begin{bmatrix} 1&0\\0&u_{ij} \end{bmatrix}. \end{align*} $$

Lemma 32. The following diagram commutes in $D_{Z/2}(\mathbb {C})$

Proof. Start with an element $a_0[a_1|\cdots |a_k]$ of internal degree zero over $U_{I}$ , $I=\{i_0<i_1<...<i_p\}$ . Note that the only terms from $\phi $ which do not vanish under $HKR_{\mathcal {O}_{-f}}$ are those where $n=0$ and $q\geq 1$ or $n=2$ and $q\geq 0$ . These are mapped to

$$ \begin{align*} &-\sum_{q\geq 1} \sum\frac{(-1)^{pq+\binom{q}{2}}}{k!q!}u_{i_0i_{-q}}a_0du_{i_{1-q}i_{-q}}^{-1}\wedge...\wedge du_{i_0i_{-1}}^{-1}\wedge da_1\wedge...\wedge da_k\\ &-\sum_{q\geq 0} \sum\frac{(-1)^{pq+\binom{q}{2}}}{k!(q+1)!}u_{i_0i_{-q}}a_0du_{i_{1-q}i_{-q}}^{-1}\wedge...\wedge du_{i_0i_{-1}}^{-1}\wedge dx_{i_0}\wedge dg_{i_0}\wedge da_1\wedge...\wedge da_k \end{align*} $$

where the second sums are over all $\{(i_{-q},...,i_{-1})|i_{-q}<i_{1-q}<...<i_{-1}<i_0\}$ . This is precisely what we get when going down first and then to the right twice.

Now let’s start with an element with precisely one factor from $\mathcal {A}^{-1}$ , $\bar {b}=b_0[b_1|\cdots |b_l\epsilon _{i_0}|\cdots |b_k]$ and first go to the right twice and then down. This time, the only terms in the definition of $\phi $ which don’t vanish under $HKR_{\mathcal {O}_{-f}}$ are those with $n=1$ and $q\geq 0$ . We get

$$ \begin{align*} \sum \frac{(-1)^{1+l+pq+\binom{q}{2}}}{k!(q+1)!}u_{i_0i_{-q}}b_0du_{i_{1-q}i_{-q}}^{-1}\wedge...\wedge du_{i_0i_{-1}}^{-1}\wedge dx_{i_0}\wedge db_1\wedge...\wedge db_k \end{align*} $$

which agrees with what we get going down and then to the right.

Finally if we start with an element with two or more factors from $\mathcal {A}^{-1}$ , then going either direction is zero. This is clear if we go down first because $HKR_{\mathcal {A}}$ vanishes on such elements by definition. For the other direction note that the only terms in the definition of $\phi $ which are non-zero on such an element are the ones where n equals the number of factors from $\mathcal {A}^{-1}$ so $n\geq 2$ . Each such term will contain two or more factors $x_i$ and will therefore vanish when we apply $HKR_{\mathcal {O}_{-f}}$ because $dx_i\wedge dx_i=0$ .

Lemma 33. There is a commutative diagram in $D_{Z/2}(\mathbb {C})$

Proof. We will divide the diagram into smaller commutative pieces where each arrow is a chain map

Note first that all the vertical arrows on the left are quasi-isomorphisms by the discussion in Section 3.4. We will see below that the horizontal arrows in the squares $(A)-(C)$ are quasi-isomorphisms and then it follows from commutativity (which is also dealt with below) that the vertical arrows on the right too are quasi-isomorphisms.

In the square labeled $(A)$ the top arrow is a quasi-isomorphism by lemma 24. The bottom arrow in this square is ${\check{C}}(\text {Ind},\alpha )$ where $(\text {Ind},\alpha )$ is the lax morphism of presheaves of dg categories from proposition 26. Commutativity of $(A)$ is precisely proposition 11. The bottom arrow in this square can be shown to be a quasi-isomorphism by filtering both complexes by Cech degree and using the mapping theorem ([Reference Mac Lane12],theorem XI.3.4).

To see that the square $(B)$ commutes, we first note that we can assume that in the diagram below, we have $(\text {Ind},\alpha )\circ Yoneda=(Yoneda\circ q,1)$ as lax morphisms of presheaves of dg categories

Commutativity then follows from Lemma 14.

Commutativity of the square $(C)$ is part of Proposition 27.

Lemma 34. There is a commutative diagram in $D_{Z/2}(\mathbb {C})$

Proof. We will divide the diagram into smaller commutative pieces where each arrow is a chain map:

We already know that the vertical arrows on the left and right are quasi-isomorphisms. Indeed, on the left hand side this was dealt with in the proof of the previous lemma. Those on the right hand side were dealt with in section 3.3.

The bottom arrow of the square $(A)$ is ${\check{C}}(-\otimes _{\mathcal {A}} P,\beta )$ where $(-\otimes _{\mathcal {A}} P,\beta )$ is the lax morphism of presheaves of dg categories from Proposition 26. Commutativity of the square (A) is exactly Proposition 11. Commutativity of (B) is similar to commutativity of the square (B) in the previous lemma. Note that all arrows except the top horizontal one are induced by morphisms of presheaves of dg categories and the top map is induced by a lax morphism. In the square $(C)$ , $\mathscr {C}$ denotes the presheaf of cdg categories $\langle P,\mathcal {O}_X\rangle \subset \underline {\text {qvect}}_{nv}(X,f)$ . All the arrows in $(C)$ are induced by morphisms of presheaves of cdg categories and the diagram commutes already at the level of presheaves of cdg categories. The same holds for the square (D). In the triangle (F) all the arrows are quasi-isomorphisms and the two diagonal arrows are quasi-inverses to each other. If we choose the diagonal arrow going up then (F) clearly commutes because it comes from a commutative diagram of presheaves of cdg categories and morphisms of such. Finally the triangle (E) commutes if we choose the diagonal arrow going down by the definition of the map $\phi $ from Section 4.5.

Now we are ready to prove that the map on Hochschild homology induced by the pushforward functor $i_{\ast }:D^b(\text {coh}Y)\to D(\text {fact}_{nv}(X,f))$ is given by first multiplying by the inverse Todd class described in section 4.4 and then applying the connecting homomorphism $\delta $ coming from the short exact sequence (8). We formulate this in a precise way in the following theorem (which is the same as theorem 1 from the introduction).

Theorem 35. Let $i:Y\hookrightarrow X$ be a smooth divisor of a smooth complex variety X. Let $f\in \mathcal {O}_X(X)$ be a global function such that $f|_Y=0$ and f vanishes on its critical locus. There is a commutative diagram in $D_{Z/2}(\mathbb {C})$

Proof. Again we break it into smaller commutative pieces.

The top left triangle commutes by Lemma 25. The top left square commutes by Lemma 33. The bottom left square commutes by Lemma 28. The bottom left triangle commutes by definition of the connecting morphism $\delta $ . The part on the right commutes by Lemmas 32 and 34.